A Nonlinear Implicit Fractional Equation with Caputo Derivative

In this paper, we study a nonlinear implicit diﬀerential equation with initial conditions. The considered problem involves the fractional Caputo derivatives under some conditions on the order. We prove an existence and uniqueness analytic result by application of Banach principle. Then, another result that deals with the existence of at least one solution is delivered and some suﬃcient conditions for this result are established by means of the ﬁxed point theorem of Schaefer. At the end, we discuss two examples to illustrate the applicability of the main results.


Introduction
e theory of differential equations of fractional order and fractional calculus is very important since they can be used in analyzing and modeling real word phenomena. Recently, several researchers are interested in the important progress of differential equations of fractional order. For more information on these works and their applications, one can consult the references [1][2][3][4][5][6][7][8][9]. In particular, research on the existence of unique solutions for fractional differential equations is of big importance since it helps physicians to better understand the behaviour of real phenomena. See, for more details, the references [10][11][12][13][14]. e motivation for this work arises from both the development of the theory of fractional calculus itself and its wide applications to various fields of science, such as physics, chemistry, biology, electromagnetism of complex media, robotics, and economics.
Much attention has been paid to the existence and uniqueness of solutions of fractional dynamical systems [15][16][17][18] due to the fact that existence is the fundamental problem and a necessary condition for considering some other properties for fractional dynamical systems, such as controllability and stability. Chai [19] provided sufficient conditions for the existence of solutions to a class of antiperiodic boundary value problems for fractional differential equations, while Sheng and Jiang [20] considered a class of initial value problems for fractional differential systems.
ere are several operators studied in the field of fractional calculus, for example, see [21][22][23][24][25][26], but the difference in this work is that the operator considered is in the sense of Caputo derivative.
Motivated by the works of Benchohra et al. [27], we will establish in this paper existence and uniqueness results of the solutions of the fractional dynamical system with Caputo fractional derivative where D α is in the sense of Caputo, f: I × R n × R n ⟶ R n is a given function, x 0 , x 0 ′ , x 0 ″ ∈ R n , A is an n × n matrix, and 1 < β < 2, 3 < α < 4, with β + 2 < α.
Rest of the paper is organised as follows: in Section 2, we recall some results and definitions which we use for the proof of our main results. In Section 3, we give and prove the main theorems of this paper, and we discuss some illustrative examples.

Preliminaries
In this section, we introduce some definitions, lemmas, and preliminaries facts which are used throughout this paper. See [7] for more information. Let |.| be a suitable norm in R n and ‖.‖ be the matrix norm. We denote by C(I, R n ) the Banach space of continuous functions from I to R n with the norm We denote by L 1 (I, R n ) the space of Lebesgue-integrable function x: I ⟶ R n with the norm with the norm Definition 1. e Riemann-Liouville integral of order α > 0 for a continuous function φ ∈ L 1 ((0, 1], R) is given by with Γ(α): � ∞ 0 e − u u α− 1 du. Definition 2. If φ ∈ C n ([0, 1], R) and n − 1 < α ≤ n, then the Caputo fractional derivative is given by Lemma 1. Let n ∈ N * and n − 1 < α < n, then the general solution of D α u(t) � 0 is given by such that c i ∈ R, i � 0, 1, 2, . . . , n − 1.

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Proof. For this proof, we use the same method in [28]. We have Now, we get for all x, y ∈ X.
Theorem 1 (Banach's fixed point theorem, see [29]). Let Ω be a nonempty closed subset of a Banach space X. en, any contraction mapping T of Ω into itself has a unique fixed point.
Theorem 2 (Schaefer's fixed point theorem, see [29]). Let X be a Banach space, and let N: X ⟶ X be a completely continuous operator. If the set E � y ∈ X: y � λ Ny for some λ ∈ (0, 1) is bounded, then N has fixed points.

Main Results
We begin this section by some results that help us for solving the problem considered in (1).

Lemma 4.
For any x ∈ X and 1 < β < 2, we have Proof. By the definition of the operator D β , we have □ Journal of Mathematics Lemma 5. Let 1 < β < 2, 3 < α < 4, and G ∈ C(I, R n ). en, we can state that the problem, has for solution the following function Proof. By applying I α to both sides of equation (16), we have and using the property established in Lemmas 2 and 3, we find that Some of the initial conditions allow us to have the result. Conversely, assume that x(t) satisfy the equation (16), then we see easily the initial conditions.
We use the fact D α I α G(t) � G(t) and D α C � 0, where C is a constant; we get Let us now transform the above problem to a fixed point one. Consider the nonlinear operator T: X ⟶ X defined by To prove the main results, we need to work with the following hypotheses: (H1) e function f defined on I × R 3n is continuous.

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Also, we consider the quantities e first main result deals with the existence of a unique solution for (1). It is based on the application of Banach fixed point theorem for contraction mappings. Proof. It is sufficient for us to prove that H is a contraction mapping.
Let (x, y) ∈ X 2 . en, we can write where and

y(t), D β y(t), h(t) + AD β y(t)).
From (H2) for each t ∈ I, we have and using Lemma 4, we have erefore, we have for each t ∈ I, On the other hand, we have Journal of Mathematics which is clear in C(I, R n ). en, with the same arguments as before, we have us, we have Since D < 1, then the operator T is contraction. Hence, by Banach's contraction principle, T has a unique fixed point which is the unique solution of problem (1). e following main result deals with the existence of at least one solution of the studied problem.

Theorem 4. Under the hypotheses (H1) and (H2), problem (1) has at least one solution u(t), t ∈ I.
Proof. Let us prove the result by considering the following steps: Continuous of T: if the proof is trivial, then it is omitted (we just apply the fact that f is continuous. Uniform boundness of T: let us take r > 0 and consider the (bounded) ball B r : � x ∈ X; ‖x‖ X ≤ r . For y ∈ B r , we can write With a simple calculus, we get where m * � sup t∈I |f(t, 0, 0, 0)|. en, we have and also we have e above two inequalities show that ‖Ty‖ X < + ∞. Consequently, T is uniformly bounded. Equicontinuity of T: we prove that, for any bounded set B r for instance, we obtain that T(B r ) is an equicontinuous set of X.
Take t 1 , t 2 ∈ [0, 1], t 1 < t 2 and consider the above (bounded) ball B r of X. So, by considering y ∈ B r , we can state that As t 2 ⟶ t 1 , the right-hand side of the above inequality tends to zero, and we have also As t 2 ⟶ t 1 , the right-hand side of the above inequality tends to zero. From a consequence of the Ascoli-Arzela's theorem, we conclude that T is completely continuous.
Let y ∈ A c . en, we have y � cTy for some 0 < c < 1. Hence, we can write From (37) and (38), we state that ‖y‖ X < ∞. e set is thus bounded.
Consequently, thanks to Schaefer fixed point theorem, we deduce that T has at least one fixed point. us, problem (1) has a solution.
□ Example 1. Let us consider the following example: Journal of Mathematics 7 where f: We take We can see clearly that the function f is continuous.

Conclusion
In this work, we consider a nonlinear implicit fractional differential equation and we use the Caputo derivative operator. We prove two theorems and an example to illustrate our results. In the first theorem, we prove the existence and uniqueness of the solution and the second theorem deals with the existence of at least one solution. e methods used are the Banach's fixed point theorem and Schaefer's fixed point theorem. Here, two Caputo derivative operators of different fractional orders were used in the considered equation and it would be relevant to generalize this idea by considering several Caputo operators of different fractional orders.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares no known conflicts of interest or personal relationships that could have appeared to influence the work reported in this paper.